11: Multirate Systems - Imperial College London Systems 11: Multirate Systems •Multirate Systems •Building blocks •Resampling Cascades •Noble Identities •Noble Identities

11: Multirate Systems


• Multirate Systems

• Building blocks

• Resampling Cascades

• Noble Identities

• Noble Identities Proof

• Upsampled z-transform

• Downsampled z-transform

• Downsampled Spectrum

• Power Spectral Density +

• Perfect Reconstruction

• Commutators

• Summary

• MATLAB routines

DSP and Digital Filters (2017-9045) Multirate: 11 – 1 / 14

Multirate Systems



• Building blocks









• Commutators

• Summary

• MATLAB routines


Multirate systems include more than one sample rate

Multirate Systems



• Building blocks









• Commutators

• Summary

• MATLAB routines



Why bother?:

• May need to change the sample ratee.g. Audio sample rates include 32, 44.1, 48, 96 kHz

Multirate Systems



• Building blocks









• Commutators

• Summary

• MATLAB routines



Why bother?:


• Can relax analog or digital filter requirementse.g. Audio DAC increases sample rate so that the reconstruction filter

can have a more gradual cutoff

Multirate Systems



• Building blocks









• Commutators

• Summary

• MATLAB routines



Why bother?:


• Can relax analog or digital filter requirementse.g. Audio DAC increases sample rate so that the reconstruction filter

can have a more gradual cutoff

• Reduce computational complexityFIR filter length ∝ fs

∆fwhere ∆f is width of transition band

Lower fs ⇒ shorter filter + fewer samples⇒computation ∝ f2s

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines


Downsample y[m] = x[Km]

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else

Example:Downsample by 3 then upsample by 4

w[n]

0

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else


w[n]

0

x[m]

0

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else


w[n]

0

x[m]

0

y[r]

0

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else


w[n]

0

x[m]

0

y[r]

0

• We use different index variables (n, m, r) for different sample rates

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else


w[n]

0

x[m]

0

y[r]

0


• Use different colours for signals at different rates (sometimes)

Building blocks



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsample v[n] =

{

u[

nK

]

K | n

0 else


w[n]

0

x[m]

0

y[r]

0


• Use different colours for signals at different rates (sometimes)

• Synchronization: all signals have a sample at n = 0.

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines


Successive downsamplers or upsamplerscan be combined

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines



Upsampling can be exactly inverted

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines




Downsampling destroys informationpermanently⇒ uninvertible

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines





Resampling can be interchangediff P and Q are coprime (surprising!)

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines






Proof: Left side: y[n] = w[

1

Qn]

= x[

PQn]

if Q | n else y[n] = 0.

[Note: a | b means “a divides into b exactly”]

Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines







1

Qn]

= x[

PQn]


Right side: v[n] = u [Pn] = x[

PQn]

if Q | Pn.


Resampling Cascades



• Building blocks









• Commutators

• Summary

• MATLAB routines







1

Qn]

= x[

PQn]


Right side: v[n] = u [Pn] = x[

PQn]

if Q | Pn.

But {Q | Pn ⇒ Q | n} iff P and Q are coprime.


Noble Identities



• Building blocks









• Commutators

• Summary

• MATLAB routines


Resamplers commute with additionand multiplication

Noble Identities



• Building blocks









• Commutators

• Summary

• MATLAB routines



Delays must be multiplied by theresampling ratio

Noble Identities



• Building blocks









• Commutators

• Summary

• MATLAB routines




Noble identities:Exchange resamplers and filters

Noble Identities



• Building blocks









• Commutators

• Summary

• MATLAB routines





Example: H(z) = h[0] + h[1]z−1 + h[2]z−2 + · · ·H(z3) = h[0] + h[1]z−3 + h[2]z−6 + · · ·

Noble Identities



• Building blocks









• Commutators

• Summary

• MATLAB routines





Corrollary

Example: H(z) = h[0] + h[1]z−1 + h[2]z−2 + · · ·H(z3) = h[0] + h[1]z−3 + h[2]z−6 + · · ·

Noble Identities Proof



• Building blocks









• Commutators

• Summary

• MATLAB routines


Define hQ[n] to be theimpulse response of H(zQ).




• Building blocks









• Commutators

• Summary

• MATLAB routines



Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1.




• Building blocks









• Commutators

• Summary

• MATLAB routines



Assume that h[r] is of length M + 1 so that hQ[n] is of length QM + 1.We know that hQ[n] = 0 except when Q | n and that h[r] = hQ[Qr].




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M

m=0hQ[Qm]x[Qr −Qm]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M

m=0hQ[Qm]x[Qr −Qm] =

∑M

m=0h[m]x[Q(r −m)]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,

Upsampled Noble Identity:




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,


We know that v[n] = 0 except when Q | n and that v[Qr] = x[r].




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]

=∑M

m=0h[m]v[n−Qm]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]

=∑M

m=0h[m]v[n−Qm]

If Q ∤ n, then v[n−Qm] = 0 ∀m so w[n] = 0 = y[n]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]

=∑M

m=0h[m]v[n−Qm]


If Q | n = Qr, then w[Qr] =∑M

m=0h[m]v[Qr −Qm]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]

=∑M

m=0h[m]v[n−Qm]



m=0h[m]v[Qr −Qm]

=∑M

m=0h[m]x[r −m] = u[r]




• Building blocks









• Commutators

• Summary

• MATLAB routines




w[r] = v[Qr] =∑QM

s=0hQ[s]x[Qr − s]

=∑M


∑M

m=0h[m]x[Q(r −m)]

=∑M

m=0h[m]u[r −m] = y[r] ,



w[n] =∑QM

s=0hQ[s]v[n− s] =

∑M

m=0hQ[Qm]v[n−Qm]

=∑M

m=0h[m]v[n−Qm]



m=0h[m]v[Qr −Qm]

=∑M

m=0h[m]x[r −m] = u[r] = y[Qr] ,

Upsampled z-transform



• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑

n s.t. K|n u[nK]z−n




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑

m u[m]z−Km




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑

m u[m]z−Km = U(zK)




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑


Spectrum: V (ejω) = U(ejKω)




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑


Spectrum: V (ejω) = U(ejKω)Spectrum is horizontally shrunk and replicated K times.




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑



Example:

-2 0 20

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑



Example:K = 3: three images of the original spectrum in all.

-2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑


Spectrum: V (ejω) = U(ejKω)Spectrum is horizontally shrunk and replicated K times.Total energy unchanged; power (= energy/sample) multiplied by 1

K


-2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑



K


Energy unchanged: 1

2π

∫∣

∣U(ejω)∣

∣

2dω = 1

2π

∫∣

∣V (ejω)∣

∣

2dω

-2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


V (z) =∑

n v[n]z−n =

∑


=∑



K

Upsampling normally followed by a LP filter to remove images.


Energy unchanged: 1

2π

∫∣

∣U(ejω)∣

∣

2dω = 1

2π

∫∣

∣V (ejω)∣

∣

2dω

-2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω

Downsampled z-transform



• Building blocks









• Commutators

• Summary

• MATLAB routines


Define cK [n] = δK|n[n]




• Building blocks









• Commutators

• Summary

• MATLAB routines


Define cK [n] = δK|n[n] =1

K

∑K−1

k=0e

j2πknK




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)

From previous slide:

XK(z) = Y (zK)




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K )




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K ) = 1

K

∑K−1

k=0X(e

−j2πk

K z1K )




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K ) = 1

K

∑K−1

k=0X(e

−j2πk

K z1K )

Frequency Spectrum:

Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K ) = 1

K

∑K−1

k=0X(e

−j2πk

K z1K )

Frequency Spectrum:

Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

= 1

K

(

X(ejω

K ) +X(ejω

K− 2π

K ) +X(ejω

K− 4π

K ) + · · ·)




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K ) = 1

K

∑K−1

k=0X(e

−j2πk

K z1K )

Frequency Spectrum:

Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

= 1

K

(

X(ejω

K ) +X(ejω

K− 2π

K ) +X(ejω

K− 4π

K ) + · · ·)

Average of K aliased versions, each expanded in ω by a factor of K .




• Building blocks









• Commutators

• Summary

• MATLAB routines



K

∑K−1

k=0e

j2πknK

Now define xK [n] =

{

x[n] K | n

0 K ∤ n= cK [n]x[n]

XK(z) =∑

n xK [n]z−n = 1

K

∑

n

∑K−1

k=0e

j2πkn

K x[n]z−n

= 1

K

∑K−1

k=0

∑

n x[n](

e−j2πk

K z)−n

= 1

K

∑K−1

k=0X(e

−j2πkK z)


XK(z) = Y (zK)

⇒ Y (z) = XK(z1K ) = 1

K

∑K−1

k=0X(e

−j2πk

K z1K )

Frequency Spectrum:

Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

= 1

K

(

X(ejω

K ) +X(ejω

K− 2π

K ) +X(ejω

K− 4π

K ) + · · ·)

Average of K aliased versions, each expanded in ω by a factor of K .Downsampling is normally preceded by a LP filter to prevent aliasing.

Downsampled Spectrum



• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:

K = 3Not quite limited to ± π

K

-2 0 20

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K

Shaded region shows aliasing -2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω

Example 2:

K = 3Energy all in π

K≤ |ω| < 2 π

K

-2 0 20

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω

Example 2:


K≤ |ω| < 2 π

K

No aliasing: , -2 0 20

0.5

1

ω-2 0 2

0

0.5

1

ω




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω

Example 2:


K≤ |ω| < 2 π

K


0.5

1

ω-2 0 2

0

0.5

1

ω

No aliasing: If all energy is in r πK≤ |ω| < (r + 1) π

Kfor some integer r




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω

Example 2:


K≤ |ω| < 2 π

K


0.5

1

ω-2 0 2

0

0.5

1

ω


Kfor some integer r

Normal case (r = 0): If all energy in 0 ≤ |ω| ≤ πK




• Building blocks









• Commutators

• Summary

• MATLAB routines


Y (ejω) = 1

K

∑K−1

k=0X(e

j(ω−2πk)K )

Example 1:


K


0.5

1

ω-2 0 2

0

0.5

1

ω

Energy decreases: 1

2π

∫∣

∣Y (ejω)∣

∣

2dω ≈ 1

K× 1

2π

∫∣

∣X(ejω)∣

∣

2dω

Example 2:


K≤ |ω| < 2 π

K


0.5

1

ω-2 0 2

0

0.5

1

ω


Kfor some integer r

Normal case (r = 0): If all energy in 0 ≤ |ω| ≤ πK

Downsampling: Total energy multiplied by ≈ 1

K(= 1

Kif no aliasing)

Average power ≈ unchanged (= energy/sample)

Power Spectral Density +



• Building blocks









• Commutators

• Summary

• MATLAB routines


Example: Signal in ω ∈ ±0.4π + Tone @ ω = ±0.1π + White noise

-3 -2 -1 0 1 2 30

0.5

1

Frequency (rad/samp)

PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

∫ ∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines



Power = Energy/sample = Average PSD

-3 -2 -1 0 1 2 30

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

∫ ∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π

−πPSD(ω)dω

-3 -2 -1 0 1 2 30

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

∫ ∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π

−πPSD(ω)dω = 0.6

-3 -2 -1 0 1 2 30

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

∫ ∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π


Component powers:Signal = 0.3, Tone = 0.2, Noise = 0.1 -3 -2 -1 0 1 2 3

0

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

∫ ∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π



0

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

Upsampling:

Same energyper second⇒ Power is÷K

-3 -2 -1 0 1 2 30

0.2

0.4


PS

D , ∫ =

0.1

3 +

0.1

8 =

0.3 upsample × 2

∫

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π



0

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

Upsampling:


-3 -2 -1 0 1 2 30

0.2

0.4


PS

D , ∫ =

0.1

3 +

0.1

8 =

0.3 upsample × 2

-3 -2 -1 0 1 2 30

0.1

0.2

0.3


PS

D , ∫ =

0.0

56 +

0.1

4 =

0.2 upsample × 3

∫

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π



0

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

Upsampling:


-3 -2 -1 0 1 2 30

0.2

0.4


PS

D , ∫ =

0.1

3 +

0.1

8 =

0.3 upsample × 2

-3 -2 -1 0 1 2 30

0.1

0.2

0.3


PS

D , ∫ =

0.0

56 +

0.1

4 =

0.2 upsample × 3

Downsampling:

Average poweris unchanged.

-3 -2 -1 0 1 2 30

0.2

0.4

0.6


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

downsample ÷ 2

∫




• Building blocks









• Commutators

• Summary

• MATLAB routines




= 1

2π

∫ π



0

0.5

1


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

original rate

Upsampling:


-3 -2 -1 0 1 2 30

0.2

0.4


PS

D , ∫ =

0.1

3 +

0.1

8 =

0.3 upsample × 2

-3 -2 -1 0 1 2 30

0.1

0.2

0.3


PS

D , ∫ =

0.0

56 +

0.1

4 =

0.2 upsample × 3

Downsampling:

Average poweris unchanged.∃ aliasing inthe ÷3 case. -3 -2 -1 0 1 2 3

0

0.2

0.4

0.6


PS

D , ∫ =

0.5

+ 0

.1 =

0.6

downsample ÷ 2

-3 -2 -1 0 1 2 30

0.2

0.4

0.6


PS

D , ∫ =

0.4

9 +

0.1

1 =

0.6 downsample ÷ 3

Perfect Reconstruction



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

f i l

- --f--i--l

b e h k

-b -ef-hi-kl

a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

- --f--i--l

b e h k

-b -ef-hi-kl

a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

b e h k

-b -ef-hi-kl

a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

-b -ef-hi-kl

a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

w[m] a d g j

ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

w[m] a d g j

y[n] ab defghijkl




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

w[m] a d g j

y[n] ab defghijkl

Input sequence x[n] is split into three streams at 1

3the sample rate:

u[m] = x[3m], v[m] = x[3m− 1], w[m] = x[3m− 2]




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

w[m] a d g j

y[n] ab defghijkl


3the sample rate:

u[m] = x[3m], v[m] = x[3m− 1], w[m] = x[3m− 2]

Following upsampling, the streams are aligned by the delays and thenadded to give:

y[n] = x[n− 2]




• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

p[n] - --f--i--l

v[m] b e h k

q[n] -b -ef-hi-kl

w[m] a d g j

y[n] ab defghijkl


3the sample rate:

u[m] = x[3m], v[m] = x[3m− 1], w[m] = x[3m− 2]

Following upsampling, the streams are aligned by the delays and thenadded to give:

y[n] = x[n− 2]

Perfect Reconstruction: output is a delayed scaled replica of the input

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

e h k l

d g j m

y[n] ab defghijkl

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

e h k l

d g j m

y[n] ab defghijkl

The combination of delays and downsamplers can be regarded as acommutator that distributes values in sequence to u, w and v.

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

e h k l

d g j m

y[n] ab defghijkl

The combination of delays and downsamplers can be regarded as acommutator that distributes values in sequence to u, w and v.Fractional delays, z−

13 and z−

23 are needed to synchronize the streams.

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

e h k l

d g j m

y[n] ab defghijkl


13 and z−


The output commutator takes values from the streams in sequence.

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

v[m+ 1

3] e h k l

w[m+ 2

3] d g j m

y[n] ab defghijkl


13 and z−


The output commutator takes values from the streams in sequence.For clarity, we omit the fractional delays and regard each terminal, ◦, asholding its value until needed.

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

v[m+ 1

3] e h k l

w[m+ 2

3] d g j m

y[n] ab defghijkl


13 and z−


The output commutator takes values from the streams in sequence.For clarity, we omit the fractional delays and regard each terminal, ◦, asholding its value until needed. Initial commutator position has zero delay.

Commutators



• Building blocks









• Commutators

• Summary

• MATLAB routines


x[n] defghijklmn

u[m] f i l

v[m] b e h k

w[m] a d g j

v[m+ 1

3] e h k l

w[m+ 2

3] d g j m

y[n] ab defghijkl


13 and z−


The output commutator takes values from the streams in sequence.For clarity, we omit the fractional delays and regard each terminal, ◦, asholding its value until needed. Initial commutator position has zero delay.

The commutator direction is against the direction of the z−1 delays.

Summary



• Building blocks









• Commutators

• Summary

• MATLAB routines


• Multirate Building Blocks

◦ Upsample: X(z)1:K→ X(zK)

Invertible, Inserts K − 1 zeros between samplesShrinks and replicates spectrumFollow by LP filter to remove images

Summary



• Building blocks









• Commutators

• Summary

• MATLAB routines





◦ Downsample: X(z)K:1→ 1

K

∑K−1

k=0X(e

−j2πk

K z1K )

Destroys information and energy, keeps every K th sampleExpands and aliasses the spectrumSpectrum is the average of K aliased expanded versionsPrecede by LP filter to prevent aliases

Summary



• Building blocks









• Commutators

• Summary

• MATLAB routines






K

∑K−1

k=0X(e

−j2πk

K z1K )


• Equivalences◦ Noble Identities: H(z)←→ H(zK)◦ Interchange P : 1 and 1 : Q iff Pand Q coprime

Summary



• Building blocks









• Commutators

• Summary

• MATLAB routines






K

∑K−1

k=0X(e

−j2πk

K z1K )



• Commutators◦ Combine delays and down/up sampling

Summary



• Building blocks









• Commutators

• Summary

• MATLAB routines






K

∑K−1

k=0X(e

−j2πk

K z1K )



• Commutators◦ Combine delays and down/up sampling

For further details see Mitra: 13.

MATLAB routines



• Building blocks









• Commutators

• Summary

• MATLAB routines


resample change sampling rate

Documents

11: Multirate Systems - Imperial College London Systems 11: Multirate Systems •Multirate Systems •Building blocks •Resampling Cascades •Noble Identities •Noble Identities